The Scretching–Planck Vacuum Density and the Yang–Mills Mass Gap Part 2

This paper uses Scretching–Planck density–phase variables to express the finite-lattice \(SU(2)\) Kogut–Susskind gap as \(ΔKS=(g^2/8a)λᵤₙᵢₜ(P_0)\). Here \(P_0\) is the vacuum density, and \(λᵤₙᵢₜ(P_0)\) is the infimum of the weighted Poincaré quotient over nonconstant, gauge-invariant test functions in the product unit-sphere metric. Double-well and one-plaquette calculations check the ground-state transformation. For the four-dimensional \(SU(2)\) Euclidean Wilson measure, a plaquette Hessian bound gives \(Ric+HessU≥(2-24β)I\). For \(β<1/12\), this yields volume-uniform Poincaré and log-Sobolev inequalities that pass to the transfer-matrix vacuum slice. A conditional Otto–Reznikoff argument extends the range to \(β<0.0908\). Explicit configurations rule out a globally positive curvature bound beyond \(β=1/3\) analytically, and numerical calculations locate an obstruction near \(β=0.233\). A Gaussian two-scale example shows how the bound deteriorates as the mass tends to zero. Lattice calculations estimate scalar-channel masses in pure \(SU(2)\) and \(SU(3)\), including an exploratory \(SU(2)\) extrapolation of \(m_{0⁺⁺}/√σ=3.96(27)\). A separate \(SU(2)\)–Higgs scan reports \(W\)-channel effective masses, and a finite-volume large-field scan finds a threshold near \(ε_*≈0.48g^2\). The uniform gap bound in physical units (Conjecture C), the Hamiltonian limit, and the four-dimensional continuum construction remain open.